Integrand size = 11, antiderivative size = 191 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=-\frac {1}{2 a^{10} x^2}+\frac {10 b}{a^{11} x}+\frac {b^2}{9 a^3 (a+b x)^9}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {18 b^2}{a^{10} (a+b x)^2}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}} \]
-1/2/a^10/x^2+10*b/a^11/x+1/9*b^2/a^3/(b*x+a)^9+3/8*b^2/a^4/(b*x+a)^8+6/7* b^2/a^5/(b*x+a)^7+5/3*b^2/a^6/(b*x+a)^6+3*b^2/a^7/(b*x+a)^5+21/4*b^2/a^8/( b*x+a)^4+28/3*b^2/a^9/(b*x+a)^3+18*b^2/a^10/(b*x+a)^2+45*b^2/a^11/(b*x+a)+ 55*b^2*ln(x)/a^12-55*b^2*ln(b*x+a)/a^12
Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {\frac {a \left (-252 a^{10}+2772 a^9 b x+78419 a^8 b^2 x^2+456291 a^7 b^3 x^3+1326204 a^6 b^4 x^4+2318316 a^5 b^5 x^5+2604294 a^4 b^6 x^6+1905750 a^3 b^7 x^7+882420 a^2 b^8 x^8+235620 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^2 (a+b x)^9}+27720 b^2 \log (x)-27720 b^2 \log (a+b x)}{504 a^{12}} \]
((a*(-252*a^10 + 2772*a^9*b*x + 78419*a^8*b^2*x^2 + 456291*a^7*b^3*x^3 + 1 326204*a^6*b^4*x^4 + 2318316*a^5*b^5*x^5 + 2604294*a^4*b^6*x^6 + 1905750*a ^3*b^7*x^7 + 882420*a^2*b^8*x^8 + 235620*a*b^9*x^9 + 27720*b^10*x^10))/(x^ 2*(a + b*x)^9) + 27720*b^2*Log[x] - 27720*b^2*Log[a + b*x])/(504*a^12)
Time = 0.37 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^3 (a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \int \left (-\frac {55 b^3}{a^{12} (a+b x)}+\frac {55 b^2}{a^{12} x}-\frac {45 b^3}{a^{11} (a+b x)^2}-\frac {10 b}{a^{11} x^2}-\frac {36 b^3}{a^{10} (a+b x)^3}+\frac {1}{a^{10} x^3}-\frac {28 b^3}{a^9 (a+b x)^4}-\frac {21 b^3}{a^8 (a+b x)^5}-\frac {15 b^3}{a^7 (a+b x)^6}-\frac {10 b^3}{a^6 (a+b x)^7}-\frac {6 b^3}{a^5 (a+b x)^8}-\frac {3 b^3}{a^4 (a+b x)^9}-\frac {b^3}{a^3 (a+b x)^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log (a+b x)}{a^{12}}+\frac {45 b^2}{a^{11} (a+b x)}+\frac {10 b}{a^{11} x}+\frac {18 b^2}{a^{10} (a+b x)^2}-\frac {1}{2 a^{10} x^2}+\frac {28 b^2}{3 a^9 (a+b x)^3}+\frac {21 b^2}{4 a^8 (a+b x)^4}+\frac {3 b^2}{a^7 (a+b x)^5}+\frac {5 b^2}{3 a^6 (a+b x)^6}+\frac {6 b^2}{7 a^5 (a+b x)^7}+\frac {3 b^2}{8 a^4 (a+b x)^8}+\frac {b^2}{9 a^3 (a+b x)^9}\) |
-1/2*1/(a^10*x^2) + (10*b)/(a^11*x) + b^2/(9*a^3*(a + b*x)^9) + (3*b^2)/(8 *a^4*(a + b*x)^8) + (6*b^2)/(7*a^5*(a + b*x)^7) + (5*b^2)/(3*a^6*(a + b*x) ^6) + (3*b^2)/(a^7*(a + b*x)^5) + (21*b^2)/(4*a^8*(a + b*x)^4) + (28*b^2)/ (3*a^9*(a + b*x)^3) + (18*b^2)/(a^10*(a + b*x)^2) + (45*b^2)/(a^11*(a + b* x)) + (55*b^2*Log[x])/a^12 - (55*b^2*Log[a + b*x])/a^12
3.3.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78
| method | result | size |
| norman | \(\frac {-\frac {1}{2 a}+\frac {11 b x}{2 a^{2}}-\frac {495 b^{3} x^{3}}{a^{4}}-\frac {2970 b^{4} x^{4}}{a^{5}}-\frac {8470 b^{5} x^{5}}{a^{6}}-\frac {28875 b^{6} x^{6}}{2 a^{7}}-\frac {31647 b^{7} x^{7}}{2 a^{8}}-\frac {11319 b^{8} x^{8}}{a^{9}}-\frac {35937 b^{9} x^{9}}{7 a^{10}}-\frac {75339 b^{10} x^{10}}{56 a^{11}}-\frac {78419 b^{11} x^{11}}{504 a^{12}}}{x^{2} \left (b x +a \right )^{9}}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}\) | \(149\) |
| risch | \(\frac {\frac {55 b^{10} x^{10}}{a^{11}}+\frac {935 b^{9} x^{9}}{2 a^{10}}+\frac {10505 b^{8} x^{8}}{6 a^{9}}+\frac {15125 b^{7} x^{7}}{4 a^{8}}+\frac {20669 b^{6} x^{6}}{4 a^{7}}+\frac {27599 b^{5} x^{5}}{6 a^{6}}+\frac {36839 b^{4} x^{4}}{14 a^{5}}+\frac {50699 b^{3} x^{3}}{56 a^{4}}+\frac {78419 b^{2} x^{2}}{504 a^{3}}+\frac {11 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{9}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}+\frac {55 b^{2} \ln \left (-x \right )}{a^{12}}\) | \(151\) |
| default | \(-\frac {1}{2 a^{10} x^{2}}+\frac {10 b}{a^{11} x}+\frac {b^{2}}{9 a^{3} \left (b x +a \right )^{9}}+\frac {3 b^{2}}{8 a^{4} \left (b x +a \right )^{8}}+\frac {6 b^{2}}{7 a^{5} \left (b x +a \right )^{7}}+\frac {5 b^{2}}{3 a^{6} \left (b x +a \right )^{6}}+\frac {3 b^{2}}{a^{7} \left (b x +a \right )^{5}}+\frac {21 b^{2}}{4 a^{8} \left (b x +a \right )^{4}}+\frac {28 b^{2}}{3 a^{9} \left (b x +a \right )^{3}}+\frac {18 b^{2}}{a^{10} \left (b x +a \right )^{2}}+\frac {45 b^{2}}{a^{11} \left (b x +a \right )}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b x +a \right )}{a^{12}}\) | \(178\) |
| parallelrisch | \(\frac {-252 a^{11}+249480 \ln \left (x \right ) x^{10} a \,b^{10}-249480 \ln \left (b x +a \right ) x^{10} a \,b^{10}+997920 \ln \left (x \right ) x^{9} a^{2} b^{9}+2328480 \ln \left (x \right ) x^{8} a^{3} b^{8}+3492720 \ln \left (x \right ) x^{7} a^{4} b^{7}+3492720 \ln \left (x \right ) x^{6} a^{5} b^{6}+2328480 \ln \left (x \right ) x^{5} a^{6} b^{5}+997920 \ln \left (x \right ) x^{4} a^{7} b^{4}+249480 \ln \left (x \right ) x^{3} a^{8} b^{3}+27720 \ln \left (x \right ) x^{2} a^{9} b^{2}-997920 \ln \left (b x +a \right ) x^{9} a^{2} b^{9}-2328480 \ln \left (b x +a \right ) x^{8} a^{3} b^{8}-3492720 \ln \left (b x +a \right ) x^{7} a^{4} b^{7}-3492720 \ln \left (b x +a \right ) x^{6} a^{5} b^{6}-2328480 \ln \left (b x +a \right ) x^{5} a^{6} b^{5}-997920 \ln \left (b x +a \right ) x^{4} a^{7} b^{4}-249480 \ln \left (b x +a \right ) x^{3} a^{8} b^{3}-27720 \ln \left (b x +a \right ) x^{2} a^{9} b^{2}-78419 b^{11} x^{11}-5704776 a^{3} x^{8} b^{8}-27720 \ln \left (b x +a \right ) x^{11} b^{11}-2587464 x^{9} a^{2} b^{9}+27720 \ln \left (x \right ) x^{11} b^{11}-7975044 a^{4} b^{7} x^{7}-7276500 a^{5} b^{6} x^{6}-4268880 a^{6} b^{5} x^{5}-1496880 a^{7} b^{4} x^{4}-249480 b^{3} a^{8} x^{3}+2772 a^{10} b x -678051 a \,x^{10} b^{10}}{504 a^{12} x^{2} \left (b x +a \right )^{9}}\) | \(413\) |
(-1/2/a+11/2*b/a^2*x-495*b^3/a^4*x^3-2970*b^4/a^5*x^4-8470*b^5/a^6*x^5-288 75/2*b^6/a^7*x^6-31647/2*b^7/a^8*x^7-11319*b^8/a^9*x^8-35937/7*b^9/a^10*x^ 9-75339/56*b^10/a^11*x^10-78419/504*b^11/a^12*x^11)/x^2/(b*x+a)^9+55*b^2*l n(x)/a^12-55*b^2*ln(b*x+a)/a^12
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (177) = 354\).
Time = 0.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11} - 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (x\right )}{504 \, {\left (a^{12} b^{9} x^{11} + 9 \, a^{13} b^{8} x^{10} + 36 \, a^{14} b^{7} x^{9} + 84 \, a^{15} b^{6} x^{8} + 126 \, a^{16} b^{5} x^{7} + 126 \, a^{17} b^{4} x^{6} + 84 \, a^{18} b^{3} x^{5} + 36 \, a^{19} b^{2} x^{4} + 9 \, a^{20} b x^{3} + a^{21} x^{2}\right )}} \]
1/504*(27720*a*b^10*x^10 + 235620*a^2*b^9*x^9 + 882420*a^3*b^8*x^8 + 19057 50*a^4*b^7*x^7 + 2604294*a^5*b^6*x^6 + 2318316*a^6*b^5*x^5 + 1326204*a^7*b ^4*x^4 + 456291*a^8*b^3*x^3 + 78419*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^11 - 27720*(b^11*x^11 + 9*a*b^10*x^10 + 36*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 12 6*a^4*b^7*x^7 + 126*a^5*b^6*x^6 + 84*a^6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a^8* b^3*x^3 + a^9*b^2*x^2)*log(b*x + a) + 27720*(b^11*x^11 + 9*a*b^10*x^10 + 3 6*a^2*b^9*x^9 + 84*a^3*b^8*x^8 + 126*a^4*b^7*x^7 + 126*a^5*b^6*x^6 + 84*a^ 6*b^5*x^5 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^3 + a^9*b^2*x^2)*log(x))/(a^12*b^ 9*x^11 + 9*a^13*b^8*x^10 + 36*a^14*b^7*x^9 + 84*a^15*b^6*x^8 + 126*a^16*b^ 5*x^7 + 126*a^17*b^4*x^6 + 84*a^18*b^3*x^5 + 36*a^19*b^2*x^4 + 9*a^20*b*x^ 3 + a^21*x^2)
Time = 0.60 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {- 252 a^{10} + 2772 a^{9} b x + 78419 a^{8} b^{2} x^{2} + 456291 a^{7} b^{3} x^{3} + 1326204 a^{6} b^{4} x^{4} + 2318316 a^{5} b^{5} x^{5} + 2604294 a^{4} b^{6} x^{6} + 1905750 a^{3} b^{7} x^{7} + 882420 a^{2} b^{8} x^{8} + 235620 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{20} x^{2} + 4536 a^{19} b x^{3} + 18144 a^{18} b^{2} x^{4} + 42336 a^{17} b^{3} x^{5} + 63504 a^{16} b^{4} x^{6} + 63504 a^{15} b^{5} x^{7} + 42336 a^{14} b^{6} x^{8} + 18144 a^{13} b^{7} x^{9} + 4536 a^{12} b^{8} x^{10} + 504 a^{11} b^{9} x^{11}} + \frac {55 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{12}} \]
(-252*a**10 + 2772*a**9*b*x + 78419*a**8*b**2*x**2 + 456291*a**7*b**3*x**3 + 1326204*a**6*b**4*x**4 + 2318316*a**5*b**5*x**5 + 2604294*a**4*b**6*x** 6 + 1905750*a**3*b**7*x**7 + 882420*a**2*b**8*x**8 + 235620*a*b**9*x**9 + 27720*b**10*x**10)/(504*a**20*x**2 + 4536*a**19*b*x**3 + 18144*a**18*b**2* x**4 + 42336*a**17*b**3*x**5 + 63504*a**16*b**4*x**6 + 63504*a**15*b**5*x* *7 + 42336*a**14*b**6*x**8 + 18144*a**13*b**7*x**9 + 4536*a**12*b**8*x**10 + 504*a**11*b**9*x**11) + 55*b**2*(log(x) - log(a/b + x))/a**12
Time = 0.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {27720 \, b^{10} x^{10} + 235620 \, a b^{9} x^{9} + 882420 \, a^{2} b^{8} x^{8} + 1905750 \, a^{3} b^{7} x^{7} + 2604294 \, a^{4} b^{6} x^{6} + 2318316 \, a^{5} b^{5} x^{5} + 1326204 \, a^{6} b^{4} x^{4} + 456291 \, a^{7} b^{3} x^{3} + 78419 \, a^{8} b^{2} x^{2} + 2772 \, a^{9} b x - 252 \, a^{10}}{504 \, {\left (a^{11} b^{9} x^{11} + 9 \, a^{12} b^{8} x^{10} + 36 \, a^{13} b^{7} x^{9} + 84 \, a^{14} b^{6} x^{8} + 126 \, a^{15} b^{5} x^{7} + 126 \, a^{16} b^{4} x^{6} + 84 \, a^{17} b^{3} x^{5} + 36 \, a^{18} b^{2} x^{4} + 9 \, a^{19} b x^{3} + a^{20} x^{2}\right )}} - \frac {55 \, b^{2} \log \left (b x + a\right )}{a^{12}} + \frac {55 \, b^{2} \log \left (x\right )}{a^{12}} \]
1/504*(27720*b^10*x^10 + 235620*a*b^9*x^9 + 882420*a^2*b^8*x^8 + 1905750*a ^3*b^7*x^7 + 2604294*a^4*b^6*x^6 + 2318316*a^5*b^5*x^5 + 1326204*a^6*b^4*x ^4 + 456291*a^7*b^3*x^3 + 78419*a^8*b^2*x^2 + 2772*a^9*b*x - 252*a^10)/(a^ 11*b^9*x^11 + 9*a^12*b^8*x^10 + 36*a^13*b^7*x^9 + 84*a^14*b^6*x^8 + 126*a^ 15*b^5*x^7 + 126*a^16*b^4*x^6 + 84*a^17*b^3*x^5 + 36*a^18*b^2*x^4 + 9*a^19 *b*x^3 + a^20*x^2) - 55*b^2*log(b*x + a)/a^12 + 55*b^2*log(x)/a^12
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=-\frac {55 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{12}} + \frac {55 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac {27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11}}{504 \, {\left (b x + a\right )}^{9} a^{12} x^{2}} \]
-55*b^2*log(abs(b*x + a))/a^12 + 55*b^2*log(abs(x))/a^12 + 1/504*(27720*a* b^10*x^10 + 235620*a^2*b^9*x^9 + 882420*a^3*b^8*x^8 + 1905750*a^4*b^7*x^7 + 2604294*a^5*b^6*x^6 + 2318316*a^6*b^5*x^5 + 1326204*a^7*b^4*x^4 + 456291 *a^8*b^3*x^3 + 78419*a^9*b^2*x^2 + 2772*a^10*b*x - 252*a^11)/((b*x + a)^9* a^12*x^2)
Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {\frac {78419\,b^2\,x^2}{504\,a^3}-\frac {1}{2\,a}+\frac {50699\,b^3\,x^3}{56\,a^4}+\frac {36839\,b^4\,x^4}{14\,a^5}+\frac {27599\,b^5\,x^5}{6\,a^6}+\frac {20669\,b^6\,x^6}{4\,a^7}+\frac {15125\,b^7\,x^7}{4\,a^8}+\frac {10505\,b^8\,x^8}{6\,a^9}+\frac {935\,b^9\,x^9}{2\,a^{10}}+\frac {55\,b^{10}\,x^{10}}{a^{11}}+\frac {11\,b\,x}{2\,a^2}}{a^9\,x^2+9\,a^8\,b\,x^3+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^5+126\,a^5\,b^4\,x^6+126\,a^4\,b^5\,x^7+84\,a^3\,b^6\,x^8+36\,a^2\,b^7\,x^9+9\,a\,b^8\,x^{10}+b^9\,x^{11}}-\frac {110\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{12}} \]
((78419*b^2*x^2)/(504*a^3) - 1/(2*a) + (50699*b^3*x^3)/(56*a^4) + (36839*b ^4*x^4)/(14*a^5) + (27599*b^5*x^5)/(6*a^6) + (20669*b^6*x^6)/(4*a^7) + (15 125*b^7*x^7)/(4*a^8) + (10505*b^8*x^8)/(6*a^9) + (935*b^9*x^9)/(2*a^10) + (55*b^10*x^10)/a^11 + (11*b*x)/(2*a^2))/(a^9*x^2 + b^9*x^11 + 9*a^8*b*x^3 + 9*a*b^8*x^10 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^5 + 126*a^5*b^4*x^6 + 126*a ^4*b^5*x^7 + 84*a^3*b^6*x^8 + 36*a^2*b^7*x^9) - (110*b^2*atanh((2*b*x)/a + 1))/a^12
Time = 0.00 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.63 \[ \int \frac {1}{x^3 (a+b x)^{10}} \, dx=\frac {1215324 a^{7} b^{4} x^{4}+2216214 a^{5} b^{6} x^{6}+2772 a^{10} b x +124740 a^{2} b^{9} x^{9}+428571 a^{8} b^{3} x^{3}+1517670 a^{4} b^{7} x^{7}+2059596 a^{6} b^{5} x^{5}+75339 a^{9} b^{2} x^{2}-2328480 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{8} x^{8}-997920 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{9} x^{9}-27720 \,\mathrm {log}\left (b x +a \right ) a^{9} b^{2} x^{2}-249480 \,\mathrm {log}\left (b x +a \right ) a^{8} b^{3} x^{3}-997920 \,\mathrm {log}\left (b x +a \right ) a^{7} b^{4} x^{4}-2328480 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{5} x^{5}-3492720 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{6} x^{6}-3492720 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{7} x^{7}-252 a^{11}-249480 \,\mathrm {log}\left (b x +a \right ) a \,b^{10} x^{10}+27720 \,\mathrm {log}\left (x \right ) a^{9} b^{2} x^{2}+249480 \,\mathrm {log}\left (x \right ) a^{8} b^{3} x^{3}+997920 \,\mathrm {log}\left (x \right ) a^{7} b^{4} x^{4}+2328480 \,\mathrm {log}\left (x \right ) a^{6} b^{5} x^{5}+3492720 \,\mathrm {log}\left (x \right ) a^{5} b^{6} x^{6}+3492720 \,\mathrm {log}\left (x \right ) a^{4} b^{7} x^{7}+2328480 \,\mathrm {log}\left (x \right ) a^{3} b^{8} x^{8}+997920 \,\mathrm {log}\left (x \right ) a^{2} b^{9} x^{9}+249480 \,\mathrm {log}\left (x \right ) a \,b^{10} x^{10}+27720 \,\mathrm {log}\left (x \right ) b^{11} x^{11}+623700 a^{3} b^{8} x^{8}-27720 \,\mathrm {log}\left (b x +a \right ) b^{11} x^{11}-3080 b^{11} x^{11}}{504 a^{12} x^{2} \left (b^{9} x^{9}+9 a \,b^{8} x^{8}+36 a^{2} b^{7} x^{7}+84 a^{3} b^{6} x^{6}+126 a^{4} b^{5} x^{5}+126 a^{5} b^{4} x^{4}+84 a^{6} b^{3} x^{3}+36 a^{7} b^{2} x^{2}+9 a^{8} b x +a^{9}\right )} \]
int(1/(x**3*(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 210*a**6*b**4*x**4 + 252*a**5*b**5*x**5 + 210*a**4*b**6*x**6 + 120*a**3* b**7*x**7 + 45*a**2*b**8*x**8 + 10*a*b**9*x**9 + b**10*x**10)),x)
( - 27720*log(a + b*x)*a**9*b**2*x**2 - 249480*log(a + b*x)*a**8*b**3*x**3 - 997920*log(a + b*x)*a**7*b**4*x**4 - 2328480*log(a + b*x)*a**6*b**5*x** 5 - 3492720*log(a + b*x)*a**5*b**6*x**6 - 3492720*log(a + b*x)*a**4*b**7*x **7 - 2328480*log(a + b*x)*a**3*b**8*x**8 - 997920*log(a + b*x)*a**2*b**9* x**9 - 249480*log(a + b*x)*a*b**10*x**10 - 27720*log(a + b*x)*b**11*x**11 + 27720*log(x)*a**9*b**2*x**2 + 249480*log(x)*a**8*b**3*x**3 + 997920*log( x)*a**7*b**4*x**4 + 2328480*log(x)*a**6*b**5*x**5 + 3492720*log(x)*a**5*b* *6*x**6 + 3492720*log(x)*a**4*b**7*x**7 + 2328480*log(x)*a**3*b**8*x**8 + 997920*log(x)*a**2*b**9*x**9 + 249480*log(x)*a*b**10*x**10 + 27720*log(x)* b**11*x**11 - 252*a**11 + 2772*a**10*b*x + 75339*a**9*b**2*x**2 + 428571*a **8*b**3*x**3 + 1215324*a**7*b**4*x**4 + 2059596*a**6*b**5*x**5 + 2216214* a**5*b**6*x**6 + 1517670*a**4*b**7*x**7 + 623700*a**3*b**8*x**8 + 124740*a **2*b**9*x**9 - 3080*b**11*x**11)/(504*a**12*x**2*(a**9 + 9*a**8*b*x + 36* a**7*b**2*x**2 + 84*a**6*b**3*x**3 + 126*a**5*b**4*x**4 + 126*a**4*b**5*x* *5 + 84*a**3*b**6*x**6 + 36*a**2*b**7*x**7 + 9*a*b**8*x**8 + b**9*x**9))